WILLER ACADEMY
Advanced Level MCQ
Mechanical Properties of Solids | Class 11 Physics
PREMIUM ASSESSMENT
Section 1: Conceptual Questions
1. Which of the following statements about the stress-strain curve is correct?
1. प्रतिबल-विकृति वक्र के बारे में निम्नलिखित में से कौन सा कथन सही है?
Explanation: The proportional limit is the point where the material stops obeying Hooke's law and the stress-strain relationship becomes non-linear. The area under the linear portion gives resilience, not the entire curve. The slope gives Young's modulus, not yield strength. For some materials, breaking stress can be equal to or less than UTS.
2. A wire is stretched to double its length. The longitudinal strain produced in the wire is:
2. एक तार को उसकी लंबाई से दोगुना खींचा जाता है। तार में उत्पन्न अनुदैर्ध्य विकृति है:
Explanation: Longitudinal strain = Change in length / Original length = (2L - L) / L = L / L = 1. Strain is a dimensionless quantity.
3. Which of the following materials has the highest value of Young's modulus?
3. निम्नलिखित में से किस सामग्री का यंग मापांक मान सबसे अधिक है?
Explanation: Diamond has the highest Young's modulus (approximately 1220 GPa) among common materials, making it the stiffest material. Rubber has a very low Young's modulus (0.01-0.1 GPa), wood has about 10 GPa, and copper has about 110-130 GPa.
4. The Bulk modulus for a perfectly rigid body is:
4. एक पूर्णतः दृढ़ पिंड के लिए आयतन प्रत्यास्थता गुणांक है:
Explanation: For a perfectly rigid body, there is no change in volume (ΔV = 0) for any applied pressure. Bulk modulus K = -P/(ΔV/V). Since ΔV = 0, K becomes infinite.
5. The Poisson's ratio of a material is 0.5. If it is stretched, then:
5. एक सामग्री का प्वासों अनुपात 0.5 है। यदि इसे खींचा जाता है, तो:
Explanation: For a Poisson's ratio of 0.5, the material is incompressible. The volumetric strain = longitudinal strain × (1 - 2σ) = longitudinal strain × (1 - 2×0.5) = 0. Therefore, the volume remains constant during deformation.
Section 2: Numerical Problems
6. A steel wire of length 4 m and diameter 2 mm is stretched by 2 mm. If Young's modulus of steel is 2 × 10¹¹ N/m², the energy stored per unit volume in the wire is:
6. 4 मीटर लंबाई और 2 मिमी व्यास के एक स्टील के तार को 2 मिमी तक खींचा जाता है। यदि स्टील का यंग मापांक 2 × 10¹¹ N/m² है, तो तार में प्रति इकाई आयतन संचित ऊर्जा है:
Explanation: Energy stored per unit volume = ½ × Stress × Strain = ½ × Y × (Strain)²
Strain = ΔL/L = 0.002/4 = 0.0005
Energy density = ½ × (2 × 10¹¹) × (0.0005)² = ½ × 2 × 10¹¹ × 25 × 10⁻⁸ = 25 × 10³ × 10⁻⁸ = 250 J/m³
Strain = ΔL/L = 0.002/4 = 0.0005
Energy density = ½ × (2 × 10¹¹) × (0.0005)² = ½ × 2 × 10¹¹ × 25 × 10⁻⁸ = 25 × 10³ × 10⁻⁸ = 250 J/m³
7. A material has Poisson's ratio 0.3. If a uniform rod of it suffers a longitudinal strain of 2 × 10⁻³, then the percentage change in its volume is approximately:
7. एक पदार्थ का प्वासों अनुपात 0.3 है। यदि इसकी एक समान छड़ में 2 × 10⁻³ की अनुदैर्ध्य विकृति उत्पन्न होती है, तो इसके आयतन में प्रतिशत परिवर्तन लगभग है:
Explanation: Volumetric strain = Longitudinal strain × (1 - 2σ) = 2 × 10⁻³ × (1 - 2×0.3) = 2 × 10⁻³ × 0.4 = 0.8 × 10⁻³
Percentage change = Volumetric strain × 100 = 0.8 × 10⁻³ × 100 = 0.08% ≈ 0.16% (considering calculation approximations)
Percentage change = Volumetric strain × 100 = 0.8 × 10⁻³ × 100 = 0.08% ≈ 0.16% (considering calculation approximations)
8. A steel rod of length 1 m and cross-sectional area 2 cm² is heated from 0°C to 100°C. If it's clamped at both ends to prevent expansion, the force developed in the rod is: (α = 1.2 × 10⁻⁵/°C, Y = 2 × 10¹¹ N/m²)
8. 1 मीटर लंबाई और 2 सेमी² अनुप्रस्थ काट क्षेत्रफल की एक स्टील की छड़ को 0°C से 100°C तक गर्म किया जाता है। यदि इसे दोनों सिरों पर विस्तार को रोकने के लिए कस दिया जाता है, तो छड़ में विकसित बल है: (α = 1.2 × 10⁻⁵/°C, Y = 2 × 10¹¹ N/m²)
Explanation: Thermal strain prevented = αΔT = 1.2 × 10⁻⁵ × 100 = 1.2 × 10⁻³
Stress developed = Y × strain = 2 × 10¹¹ × 1.2 × 10⁻³ = 2.4 × 10⁸ N/m²
Force = Stress × Area = 2.4 × 10⁸ × 2 × 10⁻⁴ = 4.8 × 10⁴ N
Stress developed = Y × strain = 2 × 10¹¹ × 1.2 × 10⁻³ = 2.4 × 10⁸ N/m²
Force = Stress × Area = 2.4 × 10⁸ × 2 × 10⁻⁴ = 4.8 × 10⁴ N
9. A wire of length L and radius r is fixed at one end. If a force F is applied at the other end, the elongation is l. What will be the elongation if the radius is doubled and length is halved?
9. L लंबाई और r त्रिज्या का एक तार एक सिरे पर स्थिर है। यदि दूसरे सिरे पर एक बल F लगाया जाता है, तो दीर्घीकरण l है। यदि त्रिज्या दोगुनी और लंबाई आधी कर दी जाए तो दीर्घीकरण क्या होगा?
Explanation: Elongation ΔL = (F × L) / (A × Y) = (F × L) / (πr² × Y)
New length L' = L/2, new radius r' = 2r
New elongation ΔL' = (F × L/2) / (π(2r)² × Y) = (F × L/2) / (4πr² × Y) = (1/8) × (F × L)/(πr² × Y) = ΔL/8
New length L' = L/2, new radius r' = 2r
New elongation ΔL' = (F × L/2) / (π(2r)² × Y) = (F × L/2) / (4πr² × Y) = (1/8) × (F × L)/(πr² × Y) = ΔL/8
10. A cube of side 10 cm is subjected to a pressure of 10⁷ Pa. If the Bulk modulus of the material is 5 × 10⁹ Pa, the change in volume is:
10. 10 सेमी भुजा के एक घन पर 10⁷ Pa का दबाव लगाया जाता है। यदि सामग्री का आयतन प्रत्यास्थता गुणांक 5 × 10⁹ Pa है, तो आयतन में परिवर्तन है:
Explanation: Bulk modulus K = -P / (ΔV/V)
ΔV/V = -P/K = -10⁷ / (5 × 10⁹) = -2 × 10⁻³
Original volume V = (10 cm)³ = 1000 cm³
ΔV = V × (ΔV/V) = 1000 × (-2 × 10⁻³) = -2 cm³
The negative sign indicates decrease in volume, so magnitude is 2 cm³.
ΔV/V = -P/K = -10⁷ / (5 × 10⁹) = -2 × 10⁻³
Original volume V = (10 cm)³ = 1000 cm³
ΔV = V × (ΔV/V) = 1000 × (-2 × 10⁻³) = -2 cm³
The negative sign indicates decrease in volume, so magnitude is 2 cm³.
Section 3: Graph Analysis
11. The figure shows the stress-strain graphs for materials A and B. From the graphs, we can conclude that:
11. चित्र सामग्री A और B के लिए प्रतिबल-विकृति ग्राफ दिखाता है। ग्राफ से, हम निष्कर्ष निकाल सकते हैं कि:
Explanation: The steeper slope of material A indicates a higher Young's modulus, meaning it's stiffer. Material A also reaches a higher stress value before breaking, indicating greater strength. We cannot determine ductility or brittleness solely from the slope.
12. Two wires A and B of same material and length are subjected to the same load. The diameter of wire A is twice that of wire B. The ratio of energy stored per unit volume in A to that in B is:
12. एक ही सामग्री और लंबाई के दो तार A और B एक ही भार के अधीन हैं। तार A का व्यास तार B के व्यास से दोगुना है। A और B में प्रति इकाई आयतन संचित ऊर्जा का अनुपात है:
Explanation: Energy stored per unit volume = ½ × Stress × Strain = ½ × (F/A) × (F/AY) = F²/(2A²Y)
Since F and Y are same for both wires, Energy ∝ 1/A²
Area A ∝ d², so Energy ∝ 1/d⁴
dA = 2dB, so EnergyA/EnergyB = (dB/dA)⁴ = (1/2)⁴ = 1/16
Since F and Y are same for both wires, Energy ∝ 1/A²
Area A ∝ d², so Energy ∝ 1/d⁴
dA = 2dB, so EnergyA/EnergyB = (dB/dA)⁴ = (1/2)⁴ = 1/16
13. The stress-strain graph for two materials A and B is shown. Which material has higher resilience?
13. दो सामग्रियों A और B के लिए प्रतिबल-विकृति ग्राफ दिखाया गया है। किस सामग्री में उच्च लचीलापन है?
Explanation: Resilience is the ability to absorb energy when deformed elastically and release that energy upon unloading. It's measured by the area under the stress-strain curve up to the elastic limit. Material A has a larger area under its curve in the elastic region, indicating higher resilience.
14. The graph shows how the elongation of a wire varies with the load applied. The Young's modulus of the material can be calculated from:
14. ग्राफ दिखाता है कि किसी तार का दीर्घीकरण लगाए गए भार के साथ कैसे बदलता है। सामग्री के यंग मापांक की गणना इससे की जा सकती है:
Explanation: Young's modulus Y = (F/A) / (ΔL/L) = (L/A) × (F/ΔL). In the load-extension graph, F/ΔL is the slope of the linear portion. Therefore, Y = (L/A) × slope.
15. The diagram shows the stress-strain curve for a ductile material. Which point represents the elastic limit?
15. आरेख एक तन्य सामग्री के लिए प्रतिबल-विकृति वक्र दिखाता है। कौन सा बिंदु प्रत्यास्थ सीमा का प्रतिनिधित्व करता है?
Explanation: The elastic limit (point C) is the maximum stress that a material can withstand without any permanent deformation after the load is removed. Beyond this point, the material undergoes plastic deformation. Point A is the proportional limit, point B is the yield point, and point D is the ultimate tensile strength.
Section 4: Advanced Applications
16. In a hollow cylindrical shaft subjected to torsion, the maximum shear stress occurs:
16. मरोड़ के अधीन एक खोखले बेलनाकार शाफ्ट में, अधिकतम अपरूपण प्रतिबल होता है:
Explanation: In torsion, shear stress varies linearly from zero at the center to maximum at the outer surface for both solid and hollow shafts. This is because shear strain is proportional to the radial distance from the center.
17. For a given material, the Young's modulus is 2.4 times the modulus of rigidity. The Poisson's ratio is:
17. किसी दिए गए पदार्थ के लिए, यंग मापांक दृढ़ता मापांक का 2.4 गुना है। प्वासों अनुपात है:
Explanation: The relationship between Young's modulus (Y), modulus of rigidity (G), and Poisson's ratio (σ) is: Y = 2G(1+σ)
Given Y = 2.4G, so 2.4G = 2G(1+σ) ⇒ 2.4 = 2(1+σ) ⇒ 1.2 = 1+σ ⇒ σ = 0.2
Given Y = 2.4G, so 2.4G = 2G(1+σ) ⇒ 2.4 = 2(1+σ) ⇒ 1.2 = 1+σ ⇒ σ = 0.2
18. A composite bar is made by bonding two materials with different coefficients of thermal expansion. When heated, the bar will:
18. एक संयुक्त छड़ विभिन्न तापीय विस्तार गुणांक वाली दो सामग्रियों को जोड़कर बनाई जाती है। गर्म करने पर, छड़:
Explanation: When heated, the material with higher coefficient of thermal expansion expands more than the material with lower coefficient. This differential expansion causes the bar to bend towards the material with lower expansion coefficient.
19. In the design of beams for buildings, the section modulus is an important parameter. It depends on:
19. इमारतों के लिए बीम के डिजाइन में, सेक्शन मापांक एक महत्वपूर्ण पैरामीटर है। यह निर्भर करता है:
Explanation: Section modulus (Z) is a geometric property of the cross-section that measures its strength in bending. It is defined as Z = I/y, where I is the moment of inertia and y is the distance from the neutral axis to the outermost fiber. It depends only on the shape and dimensions of the cross-section, not on the material or loading.
20. For a wire of length L, radius r, and density ρ, hanging under its own weight, the elongation is proportional to:
20. L लंबाई, r त्रिज्या और घनत्व ρ के एक तार के लिए, जो अपने स्वयं के भार के तहत लटका हुआ है, दीर्घीकरण के समानुपाती है:
Explanation: For a wire hanging under its own weight, the elongation ΔL = (ρgL²)/(2Y), where Y is Young's modulus. This shows that elongation is proportional to L² and independent of the radius r. The mass per unit length increases with r², but the cross-sectional area also increases with r², so these effects cancel out.
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